I don't think you have anything to worry about health-wise from lead shot, unless someone's shooting it at you.
Probably the biggest problem with shot-killed critters in general is the possibility of chomping down on a pellet and destroying a tooth or some dental work.
Shot size isn't really the determining factor when it comes to destroying edible meat; the biggest concerns should be distance and choke. If you pop a bunny at 5-10 yards with a full-choked 12 gauge, it won't matter WHAT shot size you were using -- that wabbit is gonna be shredded. Use a more open choke (Modified, or even Improved Cylinder), and if the critter breaks nearby, let him get out a ways before you take a crack at him.
I no longer use a shotgun for rabbits or tree rats (prefer a .22 pistol or rifle), but when I did, #6 shot was standard. When I've run low on shells during an action-filled hunt, I've used whatever I could find on the floorboards or jammed in the seatcracks of the truck, and under these conditions I've used BBs to #9s. I suppose if you wanted a heavier shot so it would be more likely to exit, you could go to a #4 or #5 instead of the sixes, but there really isn't a lot of difference between them on the game you'll be hunting.